Mathematical Misconceptions international roject Carlo Marchini – Working document n. 2

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Carlo Marchini Mathematics Department University of Parma

Misconceptions related with multiplication have a wide literature (Fischbein, Vergnaud, etc.). I want to point out some aspects that seem interesting, relating with the introduction of multiplication, especially if reducing it to addition. In textbooks, the product can be obtained as an iterated sum of the same addends. Even if this affirmation can be considered correct, remark that we want to introduce multiplication, not product. It is more difficult to introduce multiplication as iterated addition, speaking only of the operation and not of the addends 1. I try to give a theoretical justification of possible misconceptions relied on this approach. •

Firstly, reducing 3 × 4 to 4 + 4 + 4, we need properties of addition, in this case the associative property, with the inherent problems in this property.

•

Secondly, there is no accord whether 3 × 4 = 4 + 4 + 4 or 3 × 4 = 3 + 3 + 3 + 3. In order to equal the two interpretations of the product 3 × 4 (an instance of commutativity of multiplication) we cannot invoke to commutative property of addition.

•

Moreover the iterated sum cannot explain (e.g. in the first interpretation of product) what are 1 × 4 and 0 × 4.

These problems can disappear if we assume axiomatically the result of multiplication by 1 and by 0, and the commutative property of multiplication together with the multiplication itself, but in this manner, we need multiplication as a concept, therefore many examples of products are not enough to define multiplication as a whole. From another point of view, in the expression 3 × 4 = 4 + 4 + 4 (or 3 × 4 = 3 + 3 + 3 + 3, but in this case does not matter), the two equality members have an explicit big difference: in the first, there are two numbers, in the second one there is only one number (multiplicand). The question is: where has the multiplier disappeared? An obvious answer is that in the first equality 4 is (are?) added (with itself) three times. This answer deserves our attention: 3 is (are?) present as a number at the first member, and as a numeral adjective at the second member. It thence follows that in the equality 3 × 4 = 4 + 4 + 4 we are mixing two different languages: at the left the arithmetic language, at the right a meta-language speaking about arithmetic. 1 This task is accomplished by recursive function theory, but this theoretic setting does not look appropriate for Primary school.

Mathematical Misconceptions international roject Carlo Marchini – Working document n. 2

British Academy Grant

The multiplication reduction to addition gives to additive models a pre-eminence among models involving conceptual fields (Vergnaud). We can see this in problems such as “l’age du capitaine”: when pupils do not know which operation is needed, they use addition most frequently. Moreover, with the reduction of multiplication to addition, the distinction between addition and multiplication risks of disappearing: personally I record cases in which students are sure that the solution of equation 2x = 20 is x = 18. This mistake appears systematically and more frequently than the cases in which x = 10 is solution of 2 + x = 20. In the former cases there can be also the ‘connivance’ of the fact that with integer numbers +2 is written, simply, 2. Another effect of definition of product as iterated sum is that the meta-language involved cannot be extended to the cases of multiplications in which the multipliers aren’t natural numbers: (-3) × 4. This introduction of product gives raise to a (avoidable?, unavoidable?) misconception (Fischbein). The justification for the iterated sum model is often based on array; which, in turn, is related with Cartesian product of sets. Set-theoretical presentation of natural numbers, as (finite) sets cardinality, allows considering the multiplication as the arithmetic interpretation of Cartesian product of sets. However, remark that in this approach there are many ‘bugs’: Cartesian product is non-commutative; to get commutativity requires a developed theory of bijections. A sound definition of finite set is beyond pupils’ competencies. Moreover it is a set-theoretical operation applicable with arbitrary sets (finite or not), it is thence different from multiplication of natural numbers, an operation that we would extend, later, to integer, to rational, to real and to complex numbers. The cardinality model of multiplication hampers extensions to these richer numbers structures. Multiplication is linguistically non-commutative: in addition, we have addends and this common name is a prelude to commutative property of the operation. In multiplication, we have multiplier and multiplicand, two different names as a warning light that the two characters play different role. This happens in English and in Italian (and in other languages without Latin roots?). The difference of names is present in other ‘operations’ 2: in subtraction, we have minuend and subtrahend (remark the different verbal roots). In division, divisor and dividend. For the raising to the power (remark the lack of a single term to call this operation, a pointer of the fact that this operation appeared later

2 Commas are necessary since in natural numbers setting, subtraction and division are not defined for all pairs of numbers.

Mathematical Misconceptions international roject Carlo Marchini – Working document n. 2

British Academy Grant

(XVI – XVII century) and that it is less ‘natural’), basis (base?) and exponent. However, in all these cases of different nomenclature, excepting multiplication, the ‘operations’ are non-commutative. Textbooks writers prefer nowadays the more generic “terms of multiplication” of “factors of multiplication” instead of multiplier and multiplicand because of the disagreement between 3 + 3 + 3 + 3 and 4 + 4 + 4. This choice is preferred in view of commutative property. A difference between English and Italian is in reading 3 × 4 = 12: following English texts at my disposal, we read “three multiplied by four is twelve” (remark singular person) or “three times four make twelve” (remark plural person). In Italian is more frequent the singular “fa” (makes), “dà” (gives), “è” (is), “è uguale a” (is equal to). In the first English reading, the subject is the number “three” (the multiplicand, being the second the multiplier) ; in the second one, the subject is “times” and three is not a number but a numeral adjective (multiplier, being the second the multiplicand), and this explains why the verb is conjugate in the plural. In confirmation of that 1 × 4 = 4 can be read, “One time four makes four” (but “zero times four make zero”, therefore 0 is plural!). So how must we read a × b = c: a times b make c or a time b makes c? In Italian, the subject seems to be the operation: pupils call it “la per” (and addition “la più”), substantivizing the preposition (the adverb), with definite article, therefore the verb in conjugated in the singular. In SMS Italian slang the sign × is used instead of “per”, as preposition and in composing word: “×è” instead of “perché” (why or because). In the research of misconception, involving multiplication it seems to me useful present and comment an episode recorded by Proff. G. Cataldi, L. Manganello e S.A.Pappadà in their classes (First grade Secondary School) 3.The topic was the introduction of integer numbers and operation among them. The starting point was Italian banks tradition to denote their clients’ debts by red numbers and the Italian proverbial expression “essere al verde” (equivalent to “to be penniless” or “to be hard up”) in which is used “verde” (green) with the meaning of “nought”. The use of colours for mathematical aims goes back to Indian mathematician Brahamagupta (VII century). In this episode, children chose blue for “true” numbers or “credits”, green for zero and red for “debts”. For the sake of brevity I report here only (my) translation of what happened in classroom at the moment of introducing multiplication after the introduction of addition (and subtraction) between two coloured numbers. Later on, I will comment shortly some aspects of classroom work about addition. Operations introduction is gradually with the proposition of problematic situations. The class teachers are the first persons narrators. 3

From Iacomella A., Letizia A: & Marchini C. 2006, Il pensiero strutturale come strumento formativo nella scuola secondaria di 1° grado: aspetti educativi, contenutistici ed epistemologici, Preprint n. 447 of Mathematics Department of University of Parma, July 2006. http://www.unipr.it/arpa/urdidmat/Posters

Mathematical Misconceptions international roject Carlo Marchini – Working document n. 2

British Academy Grant

«1st step.

Taking account of blue numbers, we ask a possible answer for the following calculation:

3 4 =… Children have no problem to give the answer “blue twelve” since for they it is now an unquestionable fact that natural numbers are involved. 2nd step. We confronted children with

2 3 =… in addition, they answer easily, profiting by idea that to multiply means to iterate addition, they thence submit

2 3=2 2 2=6; not only did they suggest this answer but also did they propose as an illustration

4 7 = 7 7 7 7 = 28. As for addition, being solicited by us to use a correct nomenclature, they single out and record on their copybooks: Product of two blue numbers is a blue number Product of two numbers, a blue one and a red one is a red number. 3rd step Now children drill on what they have discovered till now, and everything is going well until a child realizes that they did not examine the last possible condition, this one of the kind:

3 2 =… A chorus suggests “it is equal to red six” and all of them settle the answer being convincing since the sum of two red number is red. We accept that product of two red numbers is a red number, without turning a hair, and they start calculating with this rule. At a certain point, we ask: 1) calculate 3 (7 5) =…, calculate (3 7) 2) calculate 3 (7 5) =…, calculate (3 7)

(3 5)= … and compare results; (3 5)=… and compare results.

… first results start coming…children expectations are the same as what happened for addition, … general bewilderment… they audit computations…but there is little left to be done… with the rule ‘red times red make red’, it is

3 (7 5) = 3 2 = 6 (3 5) = 21 15 = 36

(3 7)

and 6 is not equal to 36! Children, with astonishment, note that the rule they chosen, even if it is “beautiful”, is the cause of:

3 (7 5) = (3 7) 3 (7 5) ≠ (3 7)

(3 5) (3 5).

Therefore, they are in presence of a misbehaving operation: it distributes well over addition in first case and not in the second one. Therefore, the rule ‘red times red make red’ does not provide a multiplication having distributive law over addition behaving well in every case. That rule must be abandoned, since children search a new multiplication that they want to be distributive over addition, in the same way as the old, reassuring multiplication was with natural numbers. Then …? What have they to do? To surrender? Another possibility makes one way: someone suggests that

3 2=6

Mathematical Misconceptions international roject Carlo Marchini – Working document n. 2

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then he suggests exploring with this rule ‘red times red make blue’. At this point children are curious to know whether this new multiplication behaves well! It remains to do arithmetic exercises in order to assess the rule. Every exercise is going well, but at certain point,… there are circumstances in which it is necessary to multiply by 0! With some perplexity children suggest that whatever coloured numbers multiplied by 0 is 0; moreover “distributivity trials” continue. However, it is better to continue controls at home, in base of previous experiences, even if all controls could be unsuited to task. Now, with the teacher’s guarantee, the construction of the new multiplication is accomplished and the new operation is distributive over addition. It is time for recording on copybook: Product of two red numbers is a blue number. Any coloured number multiplied by 0 is 0.»

I want to point out the moments of passage quantity – magnitude – quantity, which can be gathered from the report drawn up by teachers. First, a linguistic remark. Each mathematical entity can have two aspects: a quantitative aspect and a qualitative one (e.g. for a set: the cardinality and set defining property). I am not sure that my knowledge of English allows me to enter deeply in this (philosophical) distinction. For example, I am not sure that I am able to translate correctly the difference between the length of a segment and the measure of this length. In Italian school mathematics, the length is a qualitative aspect, i.e. the equivalence class of all segments congruent with the give one (with a suitable congruent relation depending by a transformation group, following Felix Klein’s approach to Geometry). In this meaning length of a segment can be given without measuring. Measure asks for numbers. Therefore, to a segment we can stick a quantity, measure of its length, and a magnitude, its length. Quantity is the quantitative aspects of the segment; magnitude is the qualitative aspects of it. This linguistic specification is necessary for me, since my dictionary proposes “quantity” as translation for both “grandezza” e “quantità”. In this dichotomy appears an Aristotelian (or Kantian?) tenet. Ancient Greek mathematicians (Pythagoras, Hippasus, Archytas, Anaxagoras, Zeno) discovered problematic aspects concerning continuity, real numbers, infinity, for example segments incommensurability, and in this manner they (and others, Aristotle, Eudoxus) carried out a long cultural way concluding with: -

Substitution of potential infinity instead of the primitive concept of actual infinity (of atomists);

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Reduction of mathematical entities to the qualitative category of magnitude rather than to the category of quantity.

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Euclid’s Elements, is a text inspired by this epistemology, it has been held, two millenniums during, the supreme example of a mathematical text. It had a deep cultural influence overall Western and Mediterranean world. In the 5th Book of Elements, Euclid presents (Eudoxus’) magnitudes, which are interpreted as entities (whose nature is not specified). Among these entities, comparison and addition are possible, moreover it is possible to build multiples and submultiples magnitudes, in accordance with natural numbers.4 In the didactic sequence above, the epistemological passage quantity – magnitude – quantity is mediated by intuition of debt (red number) and structural approach to mathematics (Bourbaki). In the episode report, there are many occasions to point out this continuous passage. Introducing addition between coloured numbers, children treat blue numbers as quantities, the customary numbers, therefore these numbers are identified with natural numbers and addition for blue numbers is identified with the same operation for natural numbers. In the case of addition between red numbers, children proposed the same procedure attributing the nature of quantity to red numbers when addition is involved. Children have trouble for adding red and blue numbers, and these trouble is a symptom that the quantitative model for both kind of numbers is creaking. Children solve the question introducing a “weight” for numbers (the absolute value). In this manner, colour becomes a qualitative aspect, magnitude, and the number in itself (its absolute value) becomes the quantity associate to each coloured number. Children feel (implicitly) that addition for coloured numbers is truly a new operation since it must take in account qualitative and quantitative aspects at the same time. More complex dynamics are involved in multiplication. In the 1st step above the sentence: «Children have no problem […] since for they is now an unquestionable fact that natural numbers are involved»

indicates that children cling to their quantitative model for multiplication, perhaps because it is codified with the simple multiplication tables (in Italian: Pythagorean tables) 5, 6. Let us go to 2nd step: « We confronted children with 2 3 =… 4

This notion of magnitude has been elaborated in 19th century and it led to mathematical entities well known nowadays as vectors and vectors space on a field (or more generally, module on a ring). 5 Marchini C., 1988, ‘Sulle modalità di introduzione dei numeri naturali’, L'educazione Matematica, Anno IX - Serie II - 3 - N. 2, 77 - 96. Marchini C.,1991, ‘Tabelline … che passione!’, La Matematica e la sua Didattica, Anno 5, n. 1, 46-51. 6 Magnitudes product is ‘knotty’ and sometimes it doesn’t exist. In other cases product of two magnitude exists but it isn’t homogeneous with product factors, for example the product of two segments could be a plane extension.

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and they answer easily, profiting by idea that to multiply means to iterate addition, they thence submit

2 3=2 2 2=6; not only did they suggest this answer but also did they propose as an illustration

4 7 = 7 7 7 7 = 28.» Children’s remark that «they answer easily, profiting by idea that to multiply means to iterate addition,» can be interpreted as the reappearance of quantitative model associated with blue numbers. In the first product 3 is multiplier, in the second this role is played by 4, even if these factors are written in different position. The role of multiplier, thence is not determined by the position of factors, but by the ‘nature’ of factors. Red numbers are assumed as magnitudes, blue as quantities! We are here in the same situation as product by a scalar quantity in vector spaces, with the oddity that children’s ‘vector spaces’ are, from time to time, right vector spaces and left ones (in spite of commutative law of the product). Stating that product of a red and a blue numbers is red; children obtain a result (magnitude) homogeneous with the magnitude (red factor) and the quantity (blue factor). Step 3 shows another change of opinion: on the analogy of addition, the first attempt to define product of two red numbers gives a red number, coherently with the interpretation of red numbers as quantity. It is interesting that in teacher’s report there is «A chorus suggests » in which the indeterminate article reveal that not all children were convinced of the fact. Only a part of them suggests of extending the addition property to multiplication. In my opinion, the episode shows an example of epistemological obstacle, according to Bachelard 7, since «One knows against a previous knowledge, destroying what is badly done.» Herman Hankel’s 1867 Principle of formal properties maintenance is used as an assayer for children conjecture: they test the conjecture with distributive law of product over sum. There is no logical justification of this choice: at this moment, children are proposing a new operation and its properties are completely unknown. The name of multiplication and the sign used for it are mere accidents, but these accidents turn themselves into the substance of operation. In my opinion, Hankel’s Principle is a metaphysical script tacitly assumed by teacher when he presents the task (relaunching): «1) calculate 3 (7 5) =…, calculate (3 7) 2) calculate 3 (7 5) =…, calculate (3 7)

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(3 5)= … and compare results; (3 5)=… and compare results.»

Bachelard G., 1938, La formation de l'esprit scientifique, P.U.F, Paris.

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Even if names and signs for new operations compelled children to assume maintenance of property operations on natural numbers, there are not reasons for that. Structural test on distributivity, nevertheless, with its evidence, helps children to overcome epistemological obstacle: « Therefore the rule ‘red times red make red’ doesn’t provide a multiplication having distributive law

over addition behaving well in every case. That rule must be abandoned, since children search a new multiplication that they want to be distributive over addition, in the same way as the old, reassuring multiplication was with natural numbers. »

In this way, magnitude model for red numbers is abandoned once again, and the set of coloured numbers, blue, red and green ones, is assumed as a set of quantities, new, strange, numberquantities. At this stage only it is possible to set multiplication free from (iterated) addition. The episode shows how difficult and delicate is to overcome the misconceptions related to product as iterated sum. In the case reported, the long time devote to that, with a continue problematic situation, have been recovered, enhancing class success in subsequent activities and obtaining lesser presence of calculus mistakes. A rushed presentation of integer numbers and their operations leaves behind misconceptions, which resurface as unforeseen difficulties. Research projects Short term: introduce in first years of Primary School multiplication with the help of patterns and counting (see Marchini, C. & Morini G.: 1990, ‘Moltiplicazione e divisione in prima elementare’, Scuola Italiana Moderna, 99, n° 9, 15 Gennaio 1990, 62 – 63.)

8

together with

iterated addition. Compare the mistakes in the same exercises of pupils of the experimental sample and of a control one. Short term: repeat itinerary presented in the episode and assess the results.

8

In (Marchini and Morini, 1990) it is shown how to introduce division independently from addition. A geometrical pattern built with different shapes is interpreted as a space trip in which planets with different features are distinguished by the difference of the shape. The division is the result of the comparison of the trips of the space-ship 1, that stops at each planet, and the space-ship 2, that is faster but can only stops at planets with suitable features. The number of jumps made by the space ship 1 to reach a planet is the dividend. The number of different shapes of the pattern, the module, is the divisor; the number of jumps made by the space-ship 2 to reach the same planet in which ship 1 is stopped is the quotient. But if the planet in which the space ship 1 is stopped has not the right features for the second ship, then the crew can abandon the space ship by taking a shuttle at the suitable planet closer to the arrival. The number of jumps of the shuttle is the remainder of division. A similar approach can be used for the multiplication, and for the greatest common divisor.